\newproblem{lay:4_2_9}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.2.9}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	For the set below, either find an appropriate theorem to show that $W$ is a vector space or find a specific example to show the contrary.
	\begin{center}
		$W=\left\{\begin{pmatrix}p\\q\\r\\s\end{pmatrix}|p-3q=4s, 2p=s+5r\right\}$
	\end{center}
}{
  % Solution
	We can rewrite the two conditions for the vectors in $W$ as
	\begin{center}
		$\begin{pmatrix}1 & -3 & 0 & -4 \\ 2 & 0 & -5 & -1\end{pmatrix}\begin{pmatrix}p\\q\\r\\s\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$
	\end{center}
	So, $W$ is nothing more than the null space of the matrix $A=\begin{pmatrix}1 & -3 & 0 & -4 \\ 2 & 0 & -5 & -1\end{pmatrix}$ and consequently it is a vector
	subspace of $\mathbb{R}^4$. Since any vector subspace is a vector space, then $W$ is a vector space.
}
\useproblem{lay:4_2_9}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
